Maciej Zworski

Scattering matrix in conformal geometry

This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. The conformally invariant powers of the Laplacian arise as residues of the scattering matrix and Bransons Q-curvature in even dimensions as a limiting value. The integrated Q-curvature is shown to equal a multiple of the coefficient of the logarithmic term in the renormalized volume expansion.

Distribution of poles for scattering on the real line

Abstract The density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential. In the case of a potential with finite singularities at the endpoints of the support, asymptotic formulae for the poles are given, while in the C 0 ∞ case, an example of a potential with infinitely many scattering poles on i R is constructed. The scattering amplitude of a compactly supported potential is also characterized.

Scattering asymptotics for Riemann surfaces

In this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends. We also give Weyl asymptotics for the relative scattering phase of such a surface. The proofs are based on trace formulae analogous to those of the Euclidean odd-dimensional scattering. The main technical ingredient is a new proof of the Poisson formula (Theorem 5.7) which is applicable in the Euclidean case as well. Our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries. The previous general lower bounds or asymptotics were either nonoptimal ([25], [58], [9]), one-dimensional or radial ([65], [67] and [54], [41]1) or they required some degeneracy of the

Ergodicity of eigenfunctions for ergodic billiards

We give a simple proof of ergodicity of eigenfunctions of the Laplacian with Dirichlet boundary conditions on compact Riemannian manifolds with piecewise smooth boundaries and ergodic billiards. Examples include the “Bunimovich stadium”, the “Sinai billiard” and the generic polygonal billiard tables of Kerckhoff, Masur and Smillie.

Fast Soliton Scattering by Delta Impurities

We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

The Selberg zeta function for convex co-compact Schottky groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn+1: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|n+1) , and it gives new bounds on the number of resonances (scattering poles) of Γ\ℍn+1 . The proof of this result is based on the application of holomorphic L2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\ℍn+1 as the simplest model of quantum chaotic scattering.

SEMICLASSICAL ESTIMATES IN ASYMPTOTICALLY EUCLIDEAN SCATTERING

Abstract: We consider long range semiclassical perturbations of the Laplacian on asymptotically Euclidean manifolds. We obtain precise resolvent estimates under non-trapping assumptions. The novelty lies in a systematic use of geometric microlocal methods.

Fractal upper bounds on the density of semiclassical resonances

For semiclassical problems we establish upper bounds on the number of resonances in boxes of size

Lower bounds on the number of scattering poles

h

Soliton Splitting by External Delta Potentials

along the real axis, in terms of the dimension of the set of trapped trajectories. The proof uses second microlocalization.